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<p>We survey a number of incompleteness results in operator algebras stemming from the recent undecidability result in quantum complexity theory known as <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\operatorname {MIP}^*=\operatorname {RE}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mo>MIP</mo> <mo>∗</mo> </msup> <mo>=</mo> <mo>RE</mo> </mrow> </math></EquationSource> </InlineEquation>, the most prominent of which is a “Gödelian refutation” of the Connes Embedding Problem. We also discuss the very recent use of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\operatorname {MIP}^*=\operatorname {RE}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mo>MIP</mo> <mo>∗</mo> </msup> <mo>=</mo> <mo>RE</mo> </mrow> </math></EquationSource> </InlineEquation> in refuting the Aldous–Lyons conjecture in probability theory.</p>

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Undecidability and incompleteness in quantum information theory and operator algebras

  • 期刊论文

  • 发表日期：

    2025年5月8日

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  • Isaac Goldbring

摘要

We survey a number of incompleteness results in operator algebras stemming from the recent undecidability result in quantum complexity theory known as \(\operatorname {MIP}^*=\operatorname {RE}\) MIP ∗ = RE , the most prominent of which is a “Gödelian refutation” of the Connes Embedding Problem. We also discuss the very recent use of \(\operatorname {MIP}^*=\operatorname {RE}\) MIP ∗ = RE in refuting the Aldous–Lyons conjecture in probability theory.